MTE5881 Chap.7 Symmetry and Quantitative CBED
Symmetry and Quantitative CBED
QCBED refinement alternates a dynamical parameter-fit cycle with a geometric-distortion correction cycle. A credible starting thickness is found before structure factors and absorption are released, because thickness strongly controls oscillation frequency. Symmetry provides an experimental screen for defects, bending and alignment problems.
Convergence requires stable physical parameters and unstructured residuals, not merely a small change in a scalar objective. The source example searches a broad thickness region and identifies a particular basin, but those numbers belong to that specimen rather than a universal setting. Their transferable lesson is to compare pattern morphology across credible starts before releasing correlated structural variables.
Optimiser output must be judged with residual images, parameter stability, sensitivity and withheld regions. Simplex, parabolic searches and simulated annealing offer different balances of robustness, local efficiency and computational cost; the correct choice follows what is already known about the landscape.
What this chapter covers
- 01
Symmetry screening
- 02
Starting thickness
- 03
Parameter signatures
- 04
Distortion correction
- 05
Local minima
- 06
Optimisation choice
Worked example · free
Diagnose a local thickness basin
- +2Hold structural parameters fixed and compare calculated patterns over a plausible thickness range.
- +2Choose a basin whose oscillation morphology matches experiment and restart locally.
- +2Release structure factors and absorption only after thickness stabilises, then inspect residuals and repeated starts.
Key terms
- Thickness basin
- A region of parameter space where calculated oscillation morphology plausibly matches experiment.
- Residual map
- A spatial image of experimental minus calculated intensity used to diagnose systematic misfit.
- Simulated annealing
- A robust but computationally expensive optimisation approach for broad landscape exploration.
- Local minimum
- A parameter-space basin whose objective is lower nearby but may not be the best physical solution.
- Parameter coupling
- A condition in which multiple variables produce partly interchangeable changes in calculated intensity.
- Withheld region
- Experimental pixels excluded from optimisation and later used to test prediction outside the fitted mask.
- Iteration ledger
- A record of starts, released variables, objectives, parameter changes and residual maps across refinement cycles.
- Distortion cycle
- A refinement stage using the best calculation as a reference for correcting non-linear pattern-coordinate errors.
- Profile likelihood
- A constrained parameter scan used to expose flat directions, correlations and uncertainty hidden by one optimum.
Symmetry and Quantitative CBED FAQ
Why is thickness refined first?
Thickness strongly changes the angular frequency of dynamical intensity oscillations. If it is wrong, released structure factors may compensate and create a low score with physically distorted parameters. A morphology-based thickness search reduces that risk.
How does symmetry help the measurement?
Expected pattern symmetry can reveal defects, bending, astigmatism, misalignment or distortion before structural fitting. The captured material supports this quality-control role, while detailed symmetry definitions should follow the current lecture resources.
When is a refinement converged?
Diffraction parameters and distortion corrections should stabilise, residual images should lose systematic features, and credible nearby starts should reach compatible solutions. A slowly changing objective alone can reflect numerical stagnation rather than physical convergence.
Exam move
Learn the refinement as an ordered loop. Screen symmetry and specimen quality, prepare coordinates, search thickness with other parameters constrained, alternate physical fitting and distortion correction, release structure factors and absorption in stages, and inspect residuals on a fixed scale. Make a table linking oscillation frequency to thickness, near-Bragg shape to structure factors and magnitude to absorption or scale.
Compare optimisation methods by landscape knowledge and computational cost. For every claimed convergence, ask whether parameters are stable, residual structure is gone and a withheld pattern region is predicted. Build an iteration ledger containing starting values, bounds, released variables, objective, parameter changes and a residual-map reference.
Challenge the selected thickness basin from starts on both sides and compare spatial morphology rather than one score. Predefine the symmetry relation used to accept or reject a specimen region, then record defects, bending, alignment and detector-distortion checks. Release only variables with independent sensitivity and freeze those that trade along a flat direction.
Use the same residual colour scale between cycles so a visual improvement is real. Withhold one disc or pixel region and require the fitted model to predict it. If two starts reach incompatible minima, do not average them; widen the search, redesign the mask or obtain an independent thickness constraint. Finish by explaining why more HPC time cannot rescue non-identifiable parameters or a symmetry-broken input pattern.
Compare three optimisation scenarios: a well-located smooth basin, several credible local basins and a poorly bounded global search. Choose a method for each and state the cost or failure risk. Plot parameter values as well as objective value across alternating cycles so compensation cannot masquerade as convergence.
Test correlations by fixing one variable or using an independent measurement, then observe whether the target structure factor remains stable. Explain why a crescent residual, oscillation-frequency error and uniform scale error point to different parts of the workflow. Stop only when additional cycles reproduce the same physical solution and withheld data agree, not when the software exhausts its iterations.
